scholarly journals A Survey of Stack-Sorting Disciplines

10.37236/1693 ◽  
2003 ◽  
Vol 9 (2) ◽  
Author(s):  
Miklós Bóna
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Young Tableau ◽  
Simplicial Complexes ◽  
Permutation Patterns ◽  
Stack Sorting

We review the various ways that stacks, their variations and their combinations, have been used as sorting devices. In particular, we show that they have been a key motivator for the study of permutation patterns. We also show that they have connections to other areas in combinatorics such as Young tableau, planar graph theory, and simplicial complexes.

Effective coloration

1976 ◽  
Vol 41 (2) ◽  
pp. 469-480 ◽  
Author(s):  
Dwight R. Bean
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Recursive Function ◽  
Function Theory ◽  
Yield Information ◽  
Countably Infinite ◽  
Degrees Of Unsolvability ◽  
Infinite Planar

AbstractWe are concerned here with recursive function theory analogs of certain problems in chromatic graph theory. The motivating question for our work is: Does there exist a recursive (countably infinite) planar graph with no recursive 4-coloring? We obtain the following results: There is a 3-colorable, recursive planar graph which, for all k, has no recursive k-coloring; every decidable graph of genus p ≥ 0 has a recursive 2(x(p) − 1)-coloring, where x(p) is the least number of colors which will suffice to color any graph of genus p; for every k ≥ 3 there is a k-colorable, decidable graph with no recursive k-coloring, and if k = 3 or if k = 4 and the 4-color conjecture fails the graph is planar; there are degree preserving correspondences between k-colorings of graphs and paths through special types of trees which yield information about the degrees of unsolvability of k-colorings of graphs.

A Characterization of Almost-Planar Graphs

1996 ◽  
Vol 5 (3) ◽  
pp. 227-245 ◽  
Author(s):  
Bradley S. Gubser
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Planar Graphs ◽  
Explicit Description ◽  
Famous Result ◽  
Minor Criterion ◽  
Excluded Minor

Kuratowski's Theorem, perhaps the most famous result in graph theory, states that K5 and K3,3 are the only non-planar graphs for which both G\e, the deletion of the edge e, and G/e, the contraction of the edge e, are planar for all edges e of G. We characterize the almost-planar graphs, those non-planar graphs for which G\e or G/e is planar for all edges e of G. This paper gives two characterizations of the almost-planar graphs: an explicit description of the structure of almost-planar graphs; and an excluded minor criterion. We also give a best possible bound on the number of edges of an almost-planar graph.

Topological correlations in cellular structures and planar graph theory

1992 ◽  
Vol 69 (18) ◽  
pp. 2674-2677 ◽  
Author(s):  
C. Godrèche ◽  
I. Kostov ◽  
I. Yekutieli
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Cellular Structures ◽  
Topological Correlations

scholarly journals Finding and Counting Permutations via CSPs

Algorithmica ◽  
2021 ◽  
Author(s):  
Benjamin Aram Berendsohn ◽  
László Kozma ◽  
Dániel Marx
Keyword(s):  
Constraint Satisfaction ◽  
Polynomial Space ◽  
Permutation Patterns ◽  
Counting Problem ◽  
Exponential Time ◽  
Running Time ◽  
Algorithmic Question ◽  
Stack Sorting ◽  
Exponential Time Algorithms ◽  
New Algorithms

AbstractPermutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this area is deciding whether a given permutation of length n contains a given pattern of length k. In this work we give two new algorithms for this well-studied problem, one whose running time is $$n^{k/4 + o(k)}$$ n k / 4 + o ( k ) , and a polynomial-space algorithm whose running time is the better of $$O(1.6181^n)$$ O ( 1 . 6181 n ) and $$O(n^{k/2 + 1})$$ O ( n k / 2 + 1 ) . These results improve the earlier best bounds of $$n^{0.47k + o(k)}$$ n 0.47 k + o ( k ) and $$O(1.79^n)$$ O ( 1 . 79 n ) due to Ahal and Rabinovich (2000) resp. Bruner and Lackner (2012) and are the fastest algorithms for the problem when $$k \in \varOmega (\log {n})$$ k ∈ Ω ( log n ) . We show that both our new algorithms and the previous exponential-time algorithms in the literature can be viewed through the unifying lens of constraint-satisfaction. Our algorithms can also count, within the same running time, the number of occurrences of a pattern. We show that this result is close to optimal: solving the counting problem in time $$f(k) \cdot n^{o(k/\log {k})}$$ f ( k ) · n o ( k / log k ) would contradict the exponential-time hypothesis (ETH). For some special classes of patterns we obtain improved running times. We further prove that 3-increasing (4321-avoiding) and 3-decreasing (1234-avoiding) permutations can, in some sense, embed arbitrary permutations of almost linear length, which indicates that a sub-exponential running time is unlikely with the current techniques, even for patterns from these restricted classes.

scholarly journals Extremal $H$-Free Planar Graphs

10.37236/8255 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Yongxin Lan ◽  
Yongtang Shi ◽  
Zi-Xia Song
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Sufficient Conditions ◽  
Subcubic Graph ◽  
Stone Theorem

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory  83 (2016) 213–230], that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield  $ex_{_\mathcal{P}}(n,H)=3n-6$ for all $n\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erdős-Stone Theorem.  We then completely determine $ex_{_\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to  $K_1+tK_{r-1}$, where $t\ge2$ and $r\ge 3$ are integers. However, determining $ex_{_\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.

scholarly journals Flip-sort and combinatorial aspects of pop-stack sorting

2021 ◽  
Vol vol. 22 no. 2, Permutation... (Special issues) ◽  
Author(s):  
Andrei Asinowski ◽  
Cyril Banderier ◽  
Benjamin Hackl
Keyword(s):  
Lattice Paths ◽  
Permutation Patterns ◽  
Seminal Work ◽  
Sorting Algorithms ◽  
Stack Sorting

Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest. Comment: This v3 just updates the journal reference, according to the publisher wish

scholarly journals A note on planar Ramsey numbers for a triangle versus wheels

2012 ◽  
Vol Vol. 14 no. 2 (Graph Theory) ◽  
Author(s):  
Guofei Zhou ◽  
Yaojun Chen ◽  
Zhengke Miao ◽  
Shariefuddin Pirzada
Keyword(s):  
Graph Theory ◽  
Planar Graph ◽  
Ramsey Number ◽  
Ramsey Numbers ◽  
International Audience

Graph Theory International audience For two given graphs G and H , the planar Ramsey number P R ( G; H ) is the smallest integer n such that every planar graph F on n vertices either contains a copy of G , or its complement contains a copy of H . In this paper, we determine all planar Ramsey numbers for a triangle versus wheels.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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